Republicans have high hopes for this seat, in the western part of Illinois, where Phil Hare, a Democrat, took over in 2006. The mood here appears to be sour, much as it is for incumbent Democrats elsewhere. Mr. Hare was once an aide to Lane Evans, who left the seat. He appears to be facing his first truly serious challenge from Bobby Schilling, a Republican who has never held public office. Mr. Schilling, who owns a pizza business, has said he would have voted against the health care bill and urged leaders to treat government like a business — you don’t spend when you’re out of money.